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Zentrum fur TechnomathematikFachbereich 3 – Mathematik und Informatik
Nonlinear Inverse UnbalanceReconstruction in Rotor dynamics
Volker Dicken, Peter Maass, Ingo Menz,
Jenny Niebsch, Ronny Ramlau
Report 03–08
Berichte aus der Technomathematik
Report 03–08 July 2003
Nonlinear Inverse Unbalance Reconstruction in
Rotor Dynamics
V. Dicken, P. Maaß, I. Menz, J. Niebsch, R. Ramlau ∗
August 18, 2003
Abstract
This paper is devoted to the identification and reconstruction of un-
balance distributions in an aircraft engine rotor with a nonlinear damping
element. We have developed a rotor model that takes into account the non-
linear behavior of a squeeze film damper between the engine’s shaft and
casing for large oscillation amplitudes. Based on the Tikhonov regulariza-
tion for nonlinear ill–posed problems, we provided a three–step algorithm
that enables us to identify and reconstruct single and distributed unbal-
ances from data measured at the casing of the engine. In view of practical
capability, the algorithms were accelerated to meet the requirement of tol-
erable computation time for larger models, too. 1
Keywords: Rotor unbalance, nonlinear damping, nonlinear inverse problem,Tikhonov regularization
MSC (1991) 65R30, 73D35, 73D50
1 Introduction
Weight reduction efforts in aeroengine design lead to flexible casing structures,sensitive to vibration excitation from the main rotors. However the market de-mands the smallest possible level of vibrations transmitted to the aircrafts, in
∗Corresponding authors: Ronny Ramlau and Jenny Niebsch, Zentrum fur Technomath-
ematik, Universitat Bremen, D-28344 Bremen, Germany, email: ramlau/[email protected]
1The presented work was supported by the BMBF project ”Inverse Bestimmung von Un-
wuchtverteilungen in Flugzeugtriebwerken”
1
Figure 1: Engine interior and mechanical model with squeeze film damper
order to maximize passenger comfort. This holds especially true for the marketsegment of long range business aircrafts, where engines are generally mounteddirectly to the fuselage, and low cabin noise and vibration levels are essential inmeeting customer expectations. Lowering vibrations caused by unbalances in therotating parts may also help to increase safety and lifespan of aircrafts throughreducing material fatigue. Since low pressure turbine synchronous vibration lev-els can be reduced with standard trim balancing methods with relatively littleeffort and expense, the focus is on reducing vibrations excited by the unbalancedistribution on the high pressure (HP) rotor (compressor, turbine and coupling).The HP shaft is not accessible after the engine is assembled, which enforces theprinciple of ’right first time’ on balancing and assembly. A direct measurementof HP vibrations in the assembled engine is impossible, due to narrow space, highforces and temperature, therefore we have to rely on indirect measurements andhence have to solve an inverse problem.
First results on the inverse reconstruction of distributed rotor unbalances frommeasurements at the casing were achieved in a feasibility study by Rienackeret. all. [12] from the BMW Rolls–Royce GmbH, and in a cooperation projectbetween Rolls–Royce Deutschland and the AG Technomathematik of the Univer-sity of Bremen (part of the CERES project, [2]). Both treat the task as a linearproblem using a linear whole engine FEM model developed at the BMW Rolls–Royce Group and programmed in NASTRAN for the forward computation andthe construction of the inverse problem operator. In both cases it was possibleto reconstruct continuous unbalance distributions like bend or wave deformation,but they failed in the identification of point unbalance positions. It was onlypossible to give hints in which part of the HP rotor the unbalance was situated.One possible reason for the failure of this first attempt seemed to be the linearityof the NASTRAN model since it did not take into account nonlinear dampingeffects arising from squeeze film dampers at the bearing of the rotor (see Figure1). Hence it was our aim to develop a nonlinear model and check if it providesbetter reconstruction and identification results. Since it was to complicated tochange the NASTRAN model, we employed a phantom turbine model based onfinite element methods for flexible shafts. The parameters of the model were
2
trimmed to meet the characterizing properties of a high pressure rotor of an air-craft engine, like dimensions, eigenforms and resonance frequencies.The solution of the forward problem, i.e. the computation of the vibration ofthe engine for a given unbalance distribution can be found by solving a nonlineardifferential equation. As the forward problem is nonlinear, the inverse problemof identifying unbalances from data measured at certain sensor positions at thehousing is nonlinear, too. For its solution, we have used Tikhonov regularization.Based on our model for the rotor, we employed a three-step algorithm whichenables us to identify and reconstruct continuous as well as discontinuous un-balances like single point unbalances or combinations of point unbalances. Allreconstructions were done in presence of about 15% data error. Our approachcould also help if we are not specifically interested in the unbalance position butin a good balancing result with three given positions for balancing masses (likeusual in practise). It allows to compute balancing recommendations (mass andangle) for the admitted balancing positions.
The paper is organized as follows: In Section 2 we introduce the above mentionedphantom gas turbine. Details like the organization of the matrices appearing inthe model equation, and physically properties are carefully derived. This part isprobably more interesting for the engineers among the readers. Section 3 gives abrief introduction to inverse ill-posed problems and their solution. The Tikhonovregularization as one possible solution technique will be introduced. Section 4 isdevoted to the forward problem. Starting from the model equation, we constructa nonlinear operator equation of the form A · f = g describing our problem. Heref denotes the unbalance distribution and g the data measured at the casing ofthe engine. The solution of the forward problem, i.e. computing g from given f ,is contained in this section, too.Section 5 is concerned with the inverse problem of determining f from given(possibly noisy) data g. In particular the derivative of A and its adjoint arecomputed, which is necessary to apply Tikhonov regularization. Moreover wepresent an idea how to accelerate the minimization of the Tikhonov functional,and derive the reconstruction algorithm in this section. Finally, Section 6 containsthe results of our test computations and the final three–step algorithm arisingfrom the interpretation of the first reconstruction results. We demonstrate howsingle and distributed unbalances can be identified and reconstructed, and howthe data error influences the reconstructions. We also show how the algorithmworks for balancing suggestions if balancing positions are previously fixed.
2 Gas turbine model
In this section we want to establish a gas turbine model based on flexible shafts.Usually such a system is described by an equation of the form
Mu′′ + Su = Ω2p (1)
3
with M being the mass or inertia matrix, S the stiffness matrix, u the vector ofthe degrees of freedom, p the unbalanced load vector, and Ω the frequency. Inthe presence of damping, and assuming an unbalance f as the causing load, theequation has the form
Mu′′ +Du′ + Su = Ω2P f . (2)
Here D is the sparse damping matrix, f the vector of unbalances, and P is amatrix that assigns the forces caused by f to the degrees of freedom in the FEMmodel.
2.1 Modelling of a flexible shaft using Finite Elements
The usual technique for calculating the dynamic behaviour of flexible shafts isthe Finite Element Method (FEM). To get an equation of the type (1), the rotoris divided into similar elements (see Figure 2). For each node the chosen de-grees of freedom have to take into account all geometric boundary and transitionconditions.
x
Figure 2: Flexible shaft, finitebeam elements, degrees of free-dom
W = 1L
Lβ = 1
W = 1R
Rβ = 1
PLPR
FR
FL
x
z
RMLM
Figure 3: Finite beam elements,loads, shape functions, degrees offreedom
Each finite beam element is treated separately. The motion of the element isdescribed by shape functions (Figure 3). Their scaleable amplitudes are the
4
degrees of freedom WL, βL of the left node and WR, βR of the right one. Theyare ordered by
uTe = [WL βL WR βR]. (3)
From this ordering results the order of external forces and moments
pTe = [FL ML FR MR]. (4)
The characteristics of the beam element are given by its stiffness matrix Se, andits inertia or mass matrix Me . The influence of a continuous load is describedby the element load vector pe. The element matrices and the element load vectorare derived by an energy formulation (see [4]). As mentioned in the introductionthe model parameter are chosen such that the model is very close to the highpreasure rotor of an aircraft engine. Thus the stiffness matrix was derived as
Se =E · IaxL3
12ψ −6Lψ −12ψ −6Lψ−6Lψ L2(1 + 3ψ) 6Lψ L2(−1 + 3ψ)−12ψ 6Lψ 12ψ 6Lψ−6Lψ L2(−1 + 3ψ) −6Lψ L2(−1 + 3ψ)
. (5)
E is Young’s modulus, for steel we have E = 208GPa. For round beams the axialmoment of inertia is given by
Iax = π · d4out − d4
in
64(6)
with dout for the outer diameter and din for the inner diameter of the shaft. Thelength of the element is represented by L. The shear parameter is given by ([7])
ψ =1
1 + 12 E·Iax
L2·G·As
(7)
where G is the shear modulus
G =E
2(1 + ν)(8)
with Poisson’s ratio ν. For steel ν equals 0.28. For round beams the effectiveshear area
As = A · ks (9)
is derived from the area of the cross section
A =π
4(d2out − d2
in) (10)
and the shear coefficient [1]
ks =6(1 + ν)(1 +m2)2
(7 + 6ν)(1 +m2)2 + (20 + 12ν)m2(11)
5
where
m =dindout
. (12)
The matrix of inertia is superimposed of two parts
Me = Mµe +Mµe. (13)
The first part corresponds to the translatorial mass per length
µ = % ·A (14)
where % is the density of the material. For steel % = 7850kgm−3. We have (see[4])
Mµe =µL
840× (15)
280+28ψ+4ψ2 −L(35+7ψ+2ψ2) 140−28ψ−4ψ2 L(35−7ψ−2ψ2)
−L(35+7ψ+2ψ2) L2(7+ψ2) −L(35−7ψ−2ψ2) −L2(7−ψ2)
140−28ψ−4ψ2−L(35−7ψ−2ψ2) 280+28ψ+4ψ2 L(35+7ψ+2ψ2)
L(35−7ψ−2ψ2) −L2(7−ψ2) L(35+7ψ+2ψ2) L2(7+ψ2)
.
The second part takes respect to the rotational mass per length
µ = % · Iax,
Mµe =µ
30L× (16)
36ψ2 L(15ψ−18ψ2) −36ψ2 L(15ψ−18ψ2)
L(15ψ−18ψ2) L2(10−15ψ+9ψ2) −L(15ψ−18ψ2) L2(5−15ψ+9ψ2)
−36ψ2 −L(15ψ−18ψ2) 36ψ2 −L(15ψ−18ψ2)
L(15ψ−18ψ2) L2(5−15ψ+9ψ2) −L(15ψ−18ψ2) L2(10−15ψ+9ψ2)
.
The element load vector due to continuous load results in
pe =L
120
40 + 2ψ 20 − 2ψ−L(5 + ψ) −L(5 − ψ)20 − 2ψ 40 + 2ψL(5 − ψ) L(5 + ψ)
·(pLpR
)
(17)
where pL denotes the force per length at the left node and pR the force per lengthat the right node.
To build the system matrices S and M , the element matrices Se [see (5)], Me [see(13)-(16)] resp., are combined by superimposing the elements affecting the rightnode of the ith element matrix with the ones belonging to the left node of the(i+ 1)st element matrix (Figure 4). The load vector p is build by superimposingthe element load vectors pe [see (17)] in an analogous manner.
6
12
34
Figure 4: System matrix, superimposed element matrix
2.2 Gas turbine
The gas turbine shown in Figure 5 is modelled as a system of flexible shafts by44 finite beam elements. Each of the eight disks is modelled by one element, thecoupling between the compressor and the turbine consists of two elements. Theremaining 34 elements are distributed over the shaft.
1010
1010
10
10
15
20
20
20
110
310
585310
665
110
410
510
610
1250
210
o750
o450
o750
o600
o550
o500
o700
o450
o240
o110
o75
o170
o183
Figure 5: Flexible shafts (gas turbine)
Figure 6 shows the first three elastic eigenforms derived from this model. The
7
corresponding eigenfrequencies are 199 Hz (11940 rpm), 392 Hz (23520 rpm) and652 Hz (39120 rpm).
Length in m
Rad
ius
in m
392Hz652Hz
−0.2 −0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
−0.5
−0.3
−0.1
0.1
0.3
0.5
−0.2 −0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
−0.5
−0.3
−0.1
0.1
0.3
0.5
−0.2 −0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
−0.5
−0.3
−0.1
0.1
0.3
0.5
Eigenforms and −frequencies of the elastic shaft (free−free)
199Hz
Figure 6: Elastic eigenforms and eigenfrequencies, free–free
These free-free eigenfrequencies and eigenforms are modified by the turbine’selastic bearings and the housing which is supported elastically too. To keep thecalculation simple, also the housing is modelled as a flexible shaft (Figure 7).
S1
S1
S1
S1
375 375
S2 = 600.0 MN/m
S1 = 130.0 MN/m
o780
o788
Figure 7: Turbine, housing and elastic bearings
The bearings are assumed to be isotropic, and gyroscopic effects are neglected.Modelling the system this way, the equations of the horizontal plane are decoupledfrom the ones of the vertical plane.
8
Turbine
Housing
Figure 8: Block diagonal matrix structure, non zero off diagonal elements due toelastic bearings (black squares)
Since the forces of the turbine’s elastic bearings are proportional to the relativedisplacement of the turbine’s node and the node of the housing
FTurb = s · (wTurb − wHous), (18)
the bearing’s stiffness s appears in the row of the related turbine’s degree offreedom at the columns of both envolved nodes. Due to Newton’s principle ofactio and reactio,
FHous = −FTurb, (19)
it has to be invented in the related row of the housing’s degree of freedom too.The four coefficients make up a rectangle with positve elements s at the cornerson the diagonal and negative elements −s at the off diagonal corners. The elasticbearings of the housing are supported by a rigid founding. So the stiffnes appearson the diagonal elements only. The structure of the resulting stiffness matrix Sis shown in Figure 8.
The eigenfrequencies and eigenforms which result from the model in Figure 7are shown in Figure 9. Assuming a maximum speed of 17000 rpm (283 Hz) theturbine crosses two critical speeds at 6060 rpm (101 Hz) and at 8280 rpm (138Hz). The third critical speed at 17940 rpm (299 Hz) is not crossed but approachedby 95%. The forth eigenfrequency is found at 22980 rpm (383 Hz). It is abovethe range of operating speed. The fifth eigenfrequency at 35700 rpm (595 Hz) isdominated by movements of the housing.
2.3 Model including a nonlinear squeeze film damper
Damping elements are added to the model in the same manner like the elasticbearings. But since the forces of a damper are proportional to the relative veloctiy
9
−0.2 −0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4−0.2 −0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4−0.2 −0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4−0.2 −0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4−0.2 −0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
595Hz299Hz 383Hz
101Hz 138Hz
Figure 9: Eigenforms and eigenfrequencies
of the envolved turbine’s and housing’s nodes the related force on the turbine isgiven by
FTurb = d · (wTurb − wHous) (20)
with d being the damping coefficient. The related damping matrix from (2) isvery sparse. The only non-zero entries make up a rectangle of four entries foreach damper like the elastic bearing does for the stiffness matrix.
At the nearly undamped aircraft engines, squeeze film dampers are used to reducethe vibration. They fit for the reduction of unbalance induced vibration especiallydue to their increasing damping rate at higher vibration. Figure 11 from [5] showsa schematic of a squeeze film damper. Detailes shown in Figure 12 are taken from[6]. The damping arises from the shaft’s vibration in a narrow oil filled gap.The rolling contact bearing is used for separation of the rotating shaft from thenon rotating oil film only. It does not take any static load.The squeeze film damper is a pure damping element. The damping characteristicof a squeeze film damper is nonlinear, when the strength of the damping increasesby the level of vibration. In the theory of journal bearings parameters for ageneral description of the damping behaviour can be given in order to avoidtime wasting time step integration of the differential equations. Unfortunatelyno specific parameters can be given for squeeze film dampers. Only in the special
10
375 375
1250
o780
Turbine
Sq
uee
ze F
ilm D
amp
er
Compressor
Coupling
Figure 10: Rotor model with squeeze film damper
Figure 11: Schematic of a squeeze film damper
Figure 12: Details of a squeeze film damper
11
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80
2
4
6
8
10
12
Relative gapwidth (h/H)2
Dam
ping
fact
or
φ(h)=[ 1−(h/H)2 ] −3/2
Figure 13: Damping characteristic of a squeeze film damper
case of circular shaft orbits the equations of motion can be linearized. In thecase of infinite vibration the damping is a function of the oil’s viscosity andthe bearing’s geometry only. For higher vibration the damping coefficent of thesqueeze film damper can be calculated from Figure 13 as a multiple of the onerelated to infinte vibration [5].
3 A short summary on ill–posed inverse prob-
lems
In recent years the theory of treating nonlinear ill–posed problems was well es-tablished. For an overview see [3, 8]. For the following, let A be a linear ornonlinear operator between real Hilbert spaces. With given data g, we want tofind a solution f of the equation
Af = g. (21)
The problem is called ill–posed, if the solution f does not depend continuously onthe data g. In fact, if we only have noisy data with noise level δ, i.e. ‖gδ−g‖ ≤ δ,then (provided the inverse A−1 exists) f δ = A−1gδ might be an arbitrarily badapproximation to a solution of (21). To obtain a stable solution, one has to use socalled regularization methods. The general idea is to approximate the discontin-uous inverse operator by a family of continuous operators Tα. The regularizationparameter α has to be chosen such that limδ→0 α(δ, f δ) = 0 holds. For nonlinearoperators, equation (21) might have several solutions. Thus we choose the con-cept of a f–minimum–norm solution, i.e. we are looking for a solution closest toan a priory given function f .
12
A prominent example for a regularization method is Tikhonov regularization.The regularization operator is defined by
Tαgδ = f δα = arg min
fJα(f) (22)
with the Tikhonov functional given by
Jα(f) = ‖gδ − Af‖2 + α‖f − f‖2. (23)
For the determination of the regularization parameter α, we can use the so calledMorozov’s discrepancy principle where α is chosen s.t.
δ ≤ ‖gδ − Af δα‖2 ≤ cδ (24)
holds [13, 9]. For linear operators, the minimizer of the Tikhonov functionalcan be computed by solving a linear system. In case of a nonlinear operator,optimization methods have to be used additionally to compute a minimizer of(23). A classical approach to minimize the functional Jα(f) is the use of gradientmethods. The gradient of the Tikhonov functional is given by
1
2∇Jα(f) = A′(f)∗(Af − gδ) + α(f − f), (25)
where the linear operator A′(f)∗ is the adjoint of the derivative of A at the pointf [10].
4 The forward problem
As we consider a model with nonlinear damping provided by a squeeze filmdamper, equation (2) changes to
Mu′′ +D(u)u′ + Su = Ω2P f , (26)
where D is a function of the solution u. This is due to the fact that the damping ofa squeeze film damper depends on the amplitude of the oscillation. The dampingis modeled by (see Figure 13)
D(u) = D · ϕ(|h(u)|2) (27)
with ϕ(s) =(
1 − s
H2
)−3
2
, (28)
h(u) = uin − uout. (29)
Here H is the maximal gap width in the squeeze film damper, h is the actual gapwidth which is the difference between uin, the degree of freedom of the damperposition at the shaft, and uout, the degree of freedom of the damper position atthe housing. Clearly equation (26) is nonlinear in u, and the solution can notbe computed directly. To simplify the problem, we want to assume that we onlyhave a squeeze film damper at one bearing and constant damping at the otherone, hence
D(u) = D0 +D1 · ϕ(|h(u)|2). (30)
13
4.1 Construction of the forward operator
The solution of (26) with (27)-(30) are the degrees of freedom of every knot ofthe model. However, measurements can only be taken at certain sensor positions.We want to modify the solution operator of equation (26) in such a way that itmaps the causing unbalance vector to the oscillation vector of the knots at thesensor positions. In the following we assume that we have
• s sensors
• N sources of unbalances
• L degrees of freedom
• K frequencies Ω1, · · · ,ΩK.
The solution operator
Inserting the ansatz
u(t) = u+eiΩt + u−e−iΩt (31)
f(t) = f+eiΩt + f−e−iΩt
in (26), we get the following solution
u± = [−M + Ω−2S ± iΩ−1D0︸ ︷︷ ︸
CΩ
± iΩ−1D1︸ ︷︷ ︸
DΩ
·ϕ(|h(u)|2)]−1P f±. (32)
As h = h(u), this is a fixed–point equation for u±.
Real notation
Up to here we have used complex notation in all formulae. Now we want toshift to real notation. The main reason for the shifting is that for Tikhonovregularization, which will be used for solving the problem, a convergence analysisis only available for the real case. To that end we use the representation
u(t) = uc cos(Ωt) + ius sin(Ωt),
uc = 2<(u+),
us = −2=(u+).
This means that we only need u+. Denoting
CΩ := −M + Ω−2S + iΩ−1D0, (33)
DΩ := iΩ−1D1, (34)
14
(cf. (32)) we get
uc = 2<([CΩ +DΩϕ(|h|2)]−1P f+)
us = −2=([CΩ +DΩϕ(|h|2)]−1P f+).
With the additional abbreviation
HΩ(ϕ(|h|2)) := CΩ +DΩϕ(|h|2) (35)
and the notation
fr = (<(f+),=(f+))T ,
where the superscript T stands for the transpose, we arrive at the real notationfor the solution u of (26):
ur =
(uc
us
)
= 2
(<(HΩ(ϕ(|h|2))−1P ) −=(HΩ(ϕ(|h|2))−1P )=(HΩ(ϕ(|h|2))−1P ) <(HΩ(ϕ(|h|2))−1P )
)
fr. (36)
The measurement process
We will denote the real formulation of the measured data g ∈ CI s at frequency Ωand the real formulation of h by
gr(Ω) = (<(g(Ω)),=(g(Ω)))T ∈ IR2s, (37)
h(Ω) = (<(h(Ω)),=(h(Ω)))T ∈ IR2. (38)
Thus we have for the unbalance vector fr ∈ IR2N and ur = (uc,us)T ∈ IR2L.Further we assume that
• Q1 is the matrix that extracts u at the sensor positions:(Q1 00 Q1
)
ur = gr.
• Q2 is the matrix that extracts h from ur:(Q2 00 Q2
)
ur = h.
Inserting these definitions in (36), the forward problem can be written for a fixedfrequency Ω as
(gr(Ω)h(Ω)
)
IR2s+2
= 2
<(Q1HΩ(ϕ(|h|2))−1P ) −=(Q1HΩ(ϕ(|h|2))−1P )=(Q1HΩ(ϕ(|h|2))−1P ) <(Q1HΩ(ϕ(|h|2))−1P )<(Q2HΩ(ϕ(|h|2))−1P ) −=(Q2HΩ(ϕ(|h|2))−1P )=(Q2HΩ(ϕ(|h|2))−1P ) <(Q2HΩ(ϕ(|h|2))−1P )
fr.
(39)
15
For better readability we will use the abbreviations
ΠgAΩ(ϕ(|h|2)) := 2
(<(Q1HΩ(ϕ(|h|2))−1P ) −=(Q1HΩ(ϕ(|h|2))−1P )=(Q1HΩ(ϕ(|h|2))−1P ) <(Q1HΩ(ϕ(|h|2))−1P )
)
,
ΠhAΩ(ϕ(|h|2)) := 2
(<(Q2HΩ(ϕ(|h|2))−1P ) −=(Q2HΩ(ϕ(|h|2))−1P )=(Q2HΩ(ϕ(|h|2))−1P ) <(Q2HΩ(ϕ(|h|2))−1P )
)
,
AΩ(ϕ(|h|2)) :=
(ΠgAΩ(ϕ(|h|2))ΠhAΩ(ϕ(|h|2))
)
. (40)
We remark that for fixed h the above terms are matrices which have the followingsizes:
HΩ(ϕ(|h|2)) ∈ CI L×L
Q1 ∈ IRs×L
Q2 ∈ IR1×L (41)
P ∈ IRL×N
AΩ(ϕ(|h|2)) ∈ IR(2s+2)×2N .
We only have to store the real and imaginary parts of Q1HΩ(ϕ(|h|2))−1P ∈ IRs×N
and Q2HΩ(ϕ(|h|2))−1P ) ∈ IR1×N . An example for those matrices for a specifiedfrequency and h– sample can be found under www.math.uni-bremen.de\ ramlau\unbalancematrices.
Theorem 1 Let gr(Ω) be the real data vector containing the degrees of freedomat the sensor positions, h as defined in (29) and (38), and fr the real notationof the unbalance vector. Then the data vector gr for a given unbalance can becomputed by
(gr(Ω)h(Ω)
)
= AΩ(ϕ(|h|2)) fr, (42)
where AΩ(ϕ(|h|2)) is defined by (40).
4.2 Numerical solution of the forward problem
From now on we omit the index r denoting the real notation when no confusionis possible. Solving g(Ω) = ΠgAΩ(ϕ(|h|2)) f for given f would be easy if thecorrespondin h(f) is known. Finding h requires to solve
h(Ω) = ΠhAΩ(ϕ(|h|2))f , (43)
which is a fixed–point equation for h. (We refer to (38) for the use of h andh.) If we have found h with (43), the matrix ΠgAΩ(ϕ(|h|2)) and thus g can be
16
computed immediately. In a first attempt, we have used the classical fixed–pointiteration
hk+1 = ΠhAΩ(ϕ(|hk|2)) f
for solving (43), but found it rather time consuming, since for each hk the matricesQ2HΩ(ϕ(|hk|2))−1P have to be computed. Moreover, this process has to be donefor each frequency. As for inverse problems the forward problem has to be solvedquite often, this is not practicable. Instead, we found the following approachmore suitable:As we have seen above, AΩk
(ϕ(|h|2)) is a matrix for every given h. Thus, we canprecompute for a sample of |h|2, |h1|2, . . . , |hImax
|2, the corresponding matrices
[AΩk(ϕ(|hi|2))]i=1,...,Imax, k=1,...,K
and store them in advance. Now for a given unbalance f we only have to pick thematrix which fulfills (at least approximately)
hi = ΠhAΩk(ϕ(|hi|2))f . (44)
A promising fixed–point candidate for (44) is a zero of the real valued function
p(hi) = |hi|2 − |ΠhAΩ(ϕ(|hi|2))f |2. (45)
For finding such a zero hz of p, the function fzero of matlab was employed. Itinterpolates p and computes an approximative zero of the interpolated function.Thus the algorithm for the forward computation reads as follows:
• Given f , |h1|2, . . . , |hImax|2 and Ω
• p(hi) = |hi|2 − |ΠhAΩ(ϕ(|hi|2))f |2.• hz = fzero(p(hi)) (46)
• Compute ΠgAΩ(ϕ(|hz|2)) as Lagrange interpolation of ΠgAΩ(ϕ(|hi|2)) at hz
• g(Ω) = ΠgAΩ(ϕ(|hz|2)) · f (47)
In all our test computations, the zero of p(hi) was unique, which means that wehad |h|2 correctly identified. In general, the found zero |hz|2 is different from oursamples |hi|2, thus we have used Lagrange interpolation of order 2 to interpolatethe matrices AΩk
(ϕ(|hi|2)) and finally compute AΩk(ϕ(|hz|2)). We have imple-
mented both the classical fixed point method and the above described methodfor the solution of the forward problem. Several test computations assured usthat they both give the same results, but our approach was much faster.In view of (39) we only have to store the matrix
(<(Q1HΩk
(ϕ(|hi|2)))−1P ) =(Q1HΩk(ϕ(|hi|2))−1P )
<(Q2HΩk(ϕ(|hi|2)))−1P ) =(Q2HΩk
(ϕ(|hi|2))−1P )
)
∈ IR(s+1)×2N ,
17
(see (41)) for each Ωk, k = 1, · · · , K and each |hi|2, i = 1, · · · , Imax. Doing thisthe entire matrix to be stored is of size K ·(s+1)×2N ·Imax. Thus we can handlethe forward problem with maintainable amount of time and storage. Figure 14shows the movement in the squeeze film damper caused by a real unit unbalanceat the third disc.
0 50 100 150 200 250 3000
5
10
15
20
25
30
35
Omega in Hz
|h|2
Gapwidth in the squeeze film damper
Figure 14: Forward model: Oscillation in the squeeze film damper. We pass tworesonance frequencies at 101Hz and 138Hz, and approach a third one at 299Hz.
5 The inverse problem
Knowing the measured data gδ(Ω) at the sensor positions for each frequency Ω,the inverse problem consists in finding f with
gδ(Ω) = ΠgAΩ(ϕ(|h|2)) f (48)
and h(Ω) = ΠhAΩ(ϕ(|h|2)) f . (49)
It is well known from the theory of linear damped systems, that the reconstruc-tion of the causing unbalance from measured oscillation data is ill–posed. Thuswe have to use regularization methods. A widely used algorithm to solve ill–posedinverse problems is Tikhonov regularization which we have briefly described insection 3. Tikhonov regularization requires to use an algorithm for the minimiza-tion of the Tikhonov functional. For iterative algorithms, the gradient ∇J
α(f)
(see (25)) of the Tikhonov functional with nonlinear operator A plays a centralrole. For steepest descent methods, ∇J
α(f) is used as a search direction, and for
fixed point or Newton’s method, a function with
∇Jα(f) = 0
is searched for. According to (25), the computation of ∇Jα(f) requires the know-
ledge of the Frechet derivative A′(f) of A and its adjoint A′(f)∗. For our problem
18
we have
f = f ∈ IR2N ,
gδ =
gδ(Ω1)...
gδ(ΩK)
∈ IR2s·K ,
A = Πg
AΩ1
...AΩK
: IR2N −→ IR2s·K , (50)
A′ = Πg
A′
Ω1
...A′
ΩK
: IR2N −→ IR2s·K ,
A′(f)∗ = Πg
(A′
Ω1(f)∗ . . . A′
ΩK(f)∗
): IR2s·K −→ IR2N .
Here we have shorten the notation of AΩk(ϕ(|h|2)) to AΩk
.
5.1 Derivative of the forward operator and its adjoint
To find a formula for the derivative we have used the Implicit Function Theorem(IFT) which states the following: Provided a function y = y(x) is implicit de-fined via F (x, y) = 0, and F ′(x0, y0) exists and Fy(x0, y0) is invertible, then y isdifferentiable in x0 and
y′(x0) = −Fy(x0, y(x0))−1Fx(x0, y(x0)).
Theorem 2 Each component of the Frechet derivative A′(f) of A given in (50)is computed by
ΠgA′
Ω(f)v = ΠgAΩv − 2ϕ′(|h|2) 〈h,ΠhAΩv〉1 + 2ϕ′(|h|2)〈h,ΠhBΩf〉ΠgBΩf . (51)
Proof: We set
F (x, y) := F (f , (g,h)) =
(g
h
)
− AΩ(ϕ(|h|2)) f = 0.
To extract the derivative of AΩ(ϕ(|h|2)), we concider
g = ΠgAΩ(ϕ(|h|2)) f ,
and conclude from the IFT
∂fg = ΠgA′
Ω(ϕ(|h|2))(f)= Πg[−∂(g,h)F (f , (g,h))]−1∂fF (f , (g,h)).
19
The partial derivatives were computed as follows
∂fF = −AΩ(ϕ(|h|2)),
and
∂gF =
(IdIR2s
0IR2
)
.
For ∂hF we have to use Appendix A and the fact that
HΩ(ϕ(|h|2))′ = (CΩ +DΩϕ(|h|2))′ = DΩ 2 ϕ′(|h|2) 〈h, ·〉IR2
holds.
∂hF =
(0IR2s
IdIR2
)
− 2
<(Q1∂hHΩ(ϕ(|h|2))−1P ) −=(Q1∂hHΩ(ϕ(|h|2))−1P )=(Q1∂hHΩ(ϕ(|h|2))−1P ) <(Q1∂hHΩ(ϕ(|h|2))−1P )<(Q2∂hHΩ(ϕ(|h|2))−1P ) −=(Q2∂hHΩ(ϕ(|h|2))−1P )=(Q2∂hHΩ(ϕ(|h|2))−1P ) <(Q2∂hHΩ(ϕ(|h|2))−1P )
f
=
(0
Id
)
+ 2
<(Q1H−1Ω H ′
ΩH−1Ω P ) −=(Q1H
−1Ω H ′
ΩH−1Ω P )
=(Q1H−1Ω H ′
ΩH−1Ω P ) <(Q1H
−1Ω H ′
ΩH−1Ω P )
<(Q2H−1Ω H ′
ΩH−1Ω P ) −=(Q2H
−1Ω H ′
ΩH−1Ω P )
=(Q2H−1Ω H ′
ΩH−1Ω P ) <(Q2H
−1Ω H ′
ΩH−1Ω P )
f
=
(0
Id
)
+ 4〈h, ·〉IR2ϕ′(|h|2)
<(Q1H−1Ω DΩH
−1Ω P ) −=(Q1H
−1Ω DΩH
−1Ω P )
=(Q1H−1Ω DΩH
−1Ω P ) <(Q1H
−1Ω DΩH
−1Ω P )
<(Q2H−1Ω DΩH
−1Ω P ) −=(Q2H
−1Ω DΩH
−1Ω P )
=(Q2H−1Ω DΩH
−1Ω P ) <(Q2H
−1Ω DΩH
−1Ω P )
f .
We abbreviate
ΠgBΩ(ϕ(|h|2)) := 2
(<(Q1H
−1Ω DΩH
−1Ω P ) −=(Q1H
−1Ω DΩH
−1Ω P )
=(Q1H−1Ω DΩH
−1Ω P ) <(Q1H
−1Ω DΩH
−1Ω P )
)
,(52)
ΠhBΩ(ϕ(|h|2)) := 2
(<(Q2H
−1Ω DΩH
−1Ω P ) −=(Q2H
−1Ω DΩH
−1Ω P )
=(Q2H−1Ω DΩH
−1Ω P ) <(Q2H
−1Ω DΩH
−1Ω P )
)
,
BΩ(ϕ(|h|2)) :=
(ΠgBΩ(ϕ(|h|2))ΠhBΩ(ϕ(|h|2))
)
,
and arrive at
=⇒ ∂(g,h)F = IdIR2s+2 +
⟨(0
h
)
, ·⟩
2ϕ′(|h|2)BΩ(ϕ(|h|2)) f .
As stated in the theorem we have to determine (∂(g,h)F )−1. We use the identityfrom Appendix B and get
(∂(g,h)F )−1 = Id −
⟨(0
h
)
, ·⟩
2ϕ′(|h|2)BΩ(ϕ(|h)|2)f
1 + 2ϕ′(|h|2)〈h,ΠhBΩ(ϕ(|h|2))f〉 .
20
So we have ΠgA′
Ω(f) = Πg(∂(g,h)F )−1AΩ and
ΠgA′
Ω(f)v = ΠgAΩv − 2ϕ′(|h|2) 〈h,ΠhAΩv〉1 + 2ϕ′(|h|2)〈h,ΠhBΩf〉ΠgBΩf .
Next we want to compute the adjoint of the derivative.
Lemma 3 Each component of the adjoint A′(f)∗ of A′(f) given in (50) is com-puted by
[ΠgA′
Ω(f)]∗ = [ΠgAΩ]∗ − 2ϕ′(|h|2) [ΠhAΩ]∗h
1 + 2ϕ′(|h|2)〈ΠhBΩf ,h〉〈ΠgBΩf , ·〉. (53)
Proof: We have
〈[ΠgA′
Ω(f)]∗w,v〉= 〈w,ΠgA
′
Ω(f)v〉
= 〈w,ΠgAΩv〉 − 2ϕ′(|h|2)⟨
w,〈h,ΠhAΩv〉
1 + 2ϕ′(|h|2)〈h,ΠhBΩf〉ΠgBΩf
⟩
= 〈[ΠgAΩ]∗w,v〉 − 2ϕ′(|h|2)⟨ 〈ΠgBΩf ,w〉
1 + 2ϕ′(|h|2)〈ΠhBΩf ,h〉 [ΠhAΩ]∗h,v
⟩
.
We remark that in order to compute A′(f)∗, we also need to table the matrices
(<(Q1HΩ(ϕ′(|hi|2)−1DΩHΩ(ϕ′(|hi|2)−1P ) =(Q1HΩ(ϕ′(|hi|2)−1DΩHΩ(ϕ′(|hi|2)−1P )<(Q2HΩ(ϕ′(|hi|2)−1DΩHΩ(ϕ′(|hi|2)−1P ) =(Q2HΩ(ϕ′(|hi|2)−1DΩHΩ(ϕ′(|hi|2)−1P )
)
,
i = 1, · · · , Imax, from which we construct the matrix BΩ otherwise the computa-tion of A′(f)∗ would be impossible.If the derivative is known, gradient based methods are easily implemented.
5.2 On the minimization of the Tikhonov functional
For the numerical realization of the Tikhonov regularization the minimizer
f δα := arg minfJα(f)
has to be computed. For a linear operator A, this is simply done by solving thelinear system
(A∗A+ αI)f = A∗gδ + αf .
For nonlinear operator A this demands more. At first, the minimizer of theTikhonov functional might not be unique anymore, and there might be even onlylocal minimizers. Secondly, the Tikhonov functional is no longer convex. This
21
means in particular that classical algorithms like the steepest descent methodcan fail to converge to a global minimizer. In [10, 11] the so called TIGRA(TIkhonov–GRAdient)–algorithm was proposed. This method is a combinationof Tikhonov regularization and a gradient method for the minimization of theTikhonov functional. The basic algorithm is as follows:
• Given gδ with ‖gδ − g‖ ≤ δ
• Choose α0, q < 1,
• Repeat
k = k + 1
αk = qk
α0
Compute f δαk= arg min Jαk
(f) with the gradient method
until ‖Af δαk− gδ‖2 ≤ cδ2
Under slight restrictions to the nonlinear operator A it has been shown [11]that this algorithm reconstructs a good approximation to a f–minimum–norm–solution f ∗ of A f = g, provided α0 and q are chosen appropriately and
f − f ∗ = A′(f ∗)∗ ω
with ‖ω‖ small enough holds. In this case, an error estimate of the form
‖f δαk− f ∗‖ = O(
√δ) (54)
holds. We have used the TIGRA–algorithm for several reconstructions, but foundthe algorithm rather slow in convergence, which was mainly due to the highnumber of iterations for the gradient method required for the minimization ofthe Tikhonov functional for each αk. Thus, our focus was on a replacement ofthe gradient method by a faster algorithm. First, we want to show that everyminimizer of the Tikhonov functional can be characterized as a fixed point of themapping Φ defined in the following Lemma.
Lemma 4 Every minimizer of the Tikhonov functional fulfills the fixed pointequation
f = Φ(f) with (55)
Φ(f) := Tα(f)−1(A′(f)∗gδ + αf
). (56)
Proof: The necessary condition for a minimum of Jα(f) reads (see (25))
α(f − f) = A′(f)∗(gδ − Af) ,
22
which is equivalent to
(A′(f)∗A+ αI)f = A′(f)∗gδ + αf . (57)
According to (50), A′(f)∗ is a (2N × 2s · K)–matrix. By (42) we find that thedata g = Af are given by
Af = Πg
AΩ1(ϕ(|h|2))
...AΩK
(ϕ(|h|2))
f , (58)
with h = h(f). Looking at (50), we see that for given |h|2, A is a (2s ·K × 2N)–matrix, and A′(f)∗A(ϕ(|h|2)) is a (2N × 2N)–matrix.By the methods presented in Section 4.2 we can compute
A′(f)∗A(ϕ(|h|2)) f = Πg
(A′
Ω1(f)∗ . . . A′
ΩK(f)∗
)Πg
AΩ1(ϕ(|h|2))
...AΩK
(ϕ(|h|2))
f
for every given f = f , and thus the matrix
Tα(f) = A′(f)∗A(ϕ(|h|2)) + αI (59)
is well defined, and equation (57) is equivalent to
f = Φ(f) := Tα(f)−1(A′(f)∗gδ + αf
).
We remark that the belonging iteration fk+1 = Φ(fk) converged very fast to aminimizer of the Tikhonov functional.
5.3 Reconstruction algorithm
Applying Morozov’s discrepancy principle, our resulting algorithm reads as fol-lows:
• Given gδ with ‖gδ − g‖ ≤ δ
• Choose α0, q < 1, tol and f0, f1 with ‖f1 − f0‖2/‖f1‖2 > tol, k = 0
• Repeat
k = k + 1
αk = qk
α0
While ‖f1 − f0‖2/‖f1‖2 > tol
findA(f0) for f0 by computing the zero of (45)
f1 = (A′(f0)∗A(f0) + αkI)
−1(A′(f0)∗gδ + αkf)
end
f0 = f1
until ‖A(f0)f0 − gδ‖2 ≤ δ2
23
A reconstruction of an unbalance distribution from noisy data with the abovealgorithm needed in average less than two minutes (Matlab implementation, ona PC with 1.33 GHz AMD processor). In comparison, the TIGRA–algorithmneeded about an hour for a full reconstruction. As we will see in the followingSection, the above algorithm did always converge. A convergence analysis ofthe algorithm is not the scope of the present work but will be the topic of aforthcoming paper.
6 Reconstruction results
We have performed a large number of test computations using the following modelparameters:
• s = 5 sensor positions at the housing [squeeze film damper(1), bearing withconstant damping(5), and (2,3,4) equidistantly distributed between (1) and(5)]
• N = 10 sources of unbalances (compressor: eccentricities at the discs 1 to 6,turbine: eccentricities at the discs 7 and 8, coupling: run out 9, and swash10)
• K = 60 frequencies [5, 10, 15, . . . , 300]Hz
• a sample of Imax = 30 values for |h|2/H2:[0, 0.01, 0.02, . . . , 0.08, 0.1, 0.15, 0.2, . . . , 0.95, 0.99, 0.999, 0.9999]
Given an unbalance or an unbalance distribution, the data gδ were produced byforward computation and disturbed with a random additive error of 5% and arandom multiplicative error of 15%. For f we have always chosen the zero vector.
6.1 Reconstruction of a single unbalance
First, we have tested our program for several single given unbalances f , namelyfor an eccentricity at the 2nd disc in the compressor (f(2) = 0.05mm), one atthe 7th disc in the turbine (f(7) = 0.05mm), and a run out of the coupling(f(9) = 0.02mm).
6.1.1 First reconstruction results
We have plotted our reconstruction results in the Figures 15 to 17. The firstpicture (top left) always shows the reference unbalance f , the second (top right)its vibration impact at the damper sensor position. In the third picture (lower
24
0 5 10 15 200
0.01
0.02
0.03
0.04
0.05Given unbalance
0 5 10 15 20−0.01
0
0.01
0.02Reconstruction
0 100 200 3000
50
100
150Oscillation in the damper
Frequency in Hz
|h|2
0 100 200 3000
0.2
0.4
0.6
0.8Oscillation after balancing
Frequency in Hz
|h|2
Figure 15: Unbalance at 2nd disc (compressor): reconstruction results
left) the reconstructed unbalance distribution frec is plotted, and finally picturefour (lower right) shows the vibration at the damper sensor position after balanc-ing, i.e. we imagined to place balancing masses according to the reconstructeddistribution, meaning we did a forward computation with f − frec. Althoughthe results give a hint where the unbalances are situated, the algorithm fails toreconstruct the original unbalance. The reason lies in the following facts. Due tothe nature of the measurement process we are working with incomplete data. Itis a well known fact from the linear theory of rotor dynamic systems that mostinformation is contained in the resonance frequencies. There are the same num-ber of resonance frequencies as the number of knots in the model, but usuallyonly first the ten to fifteen of them are physically reasonable. Using a frequencyinterval containing all reasonable resonance frequencies, we expect the inverseproblem to have a unique solution. However, in practice it is impossible to ob-tain measurements for all reasonable resonance frequencies and we had to restrictourselves to an interval that contained two and approaches a third resonance fre-quency. Different unbalance distributions may produce different vibrations at thesensors within an interval containing all resonance frequencies, but may coincideon the considered subinterval. Thus we have lost the uniqueness of the solutionof the inverse problem. Tikhonov regularization with a given f reconstructs thena solution closest to f . In our case with f = 0, this means that we end up with areconstruction for the cause of the unbalance with minimal mass, as can be seenin the Figures 15 to 17. Although the balancing results are very good, in practicewe want to have as less balancing positions as possible due to the costs of thebalancing process. Hence the computed reconstructions are not satisfying. Toachieve a reconstruction that coincides with the given unbalance we want to locate
25
0 5 10 15 200
0.01
0.02
0.03
0.04
0.05Given unbalance
0 5 10 15 20−6
−4
−2
0
2
4x 10
−3 Reconstruction
0 100 200 3000
5
10
15
20Oscillation in the damper
Frequency in Hz
|h|2
0 100 200 3000
0.05
0.1
0.15
0.2Oscillation after balancing
Frequency in Hz
|h|2
Figure 16: Unbalance at 7th disc (turbine): reconstruction results
0 5 10 15 200
0.005
0.01
0.015
0.02Given unbalance
0 5 10 15 20−4
−2
0
2
4
6x 10
−3 Reconstruction
0 100 200 3000
2
4
6
8
10Oscillation in the damper
Frequency in Hz
|h|2
0 100 200 3000
0.01
0.02
0.03
0.04
0.05Oscillation after balancing
Frequency in Hz
|h|2
Figure 17: Run out in the coupling: reconstruction results
26
0 5 10 1510
1
102
103
104
Iteration number
2nd disc unbalance
CompressorTurbineCoupling
0 5 10 1510
1
102
103
Iteration number
7th disc unbalance
CompressorTurbineCoupling
0 5 10 1510
0
101
102
103
Iteration number
Run out(9) unbalance
CompressorTurbineCoupling
delta2 delta2 delta2
Figure 18: Residua ‖Aifδα− gδ‖2 for compressor, turbine and coupling part, indi-
cating the location of a causing unbalance
its position first. To this end we have defined an operator A[i1,...,im], im ∈ [1, N ],which is a restriction of A to the unbalance positions indicated by i1, . . . , im,i.e. this operator allows only an unbalance reconstruction at the given positions.These operators can be obtained by extracting the corresponding columns of thematrices A in (50).
6.1.2 Locating of unbalances in compressor, turbine and coupling
As stated above we want to identify the part of the engine (compressor = [1, . . . , 6],turbine = [7, 8] or coupling = [9, 10]) where the causing unbalance is located.Since we assume that we have single unbalance causes, we can only expect rea-sonable reconstructions with one of the operators Acomp, Aturb and Acoup. Accord-ing to the theory of Tikhonov regularization, a necessary condition for a goodreconstruction is that the residual ‖Af δα − gδ‖ is of the same magnitude as thedata error δ. As we can see in Figure 18, we equal the data error level only withone of the restricted operators while the residua of the others stagnated on ahigher level, and conclude that the unbalance can be located only in the part ofthe engine where the residual with the restricted operator approaches δ.
6.1.3 Locating of a single unbalance position
Once the unbalance causing part of the engine is located, we scan through thepossible unbalance positions in order to find the positions where the residuum‖A[i]f
δα,[i] − gδ‖ equals δ. For example, for the 2nd disc unbalance we did recon-
structions with A[1], · · · , A[6]. Only by using A[2] the residuum equals δ, indicatingthat the unbalance position has been found. With the belonging reconstruction
27
0 5 10 1510
1
102
103
104
Iteration number
2nd disc unbalance
123456
0 5 10 1510
1
102
103
Iteration number
7th disc unbalance
Disc 7Disc 8
0 5 10 1510
0
101
102
103
Iteration number
Run out(9) unbalance
Run outSwash
0 5 10 15 20−0.02
0
0.02
0.04
0.06unbalancereconstr.
0 5 10 15 200
0.01
0.02
0.03
0.04
0.05unbalancereconstr.
0 5 10 15 20−0.01
0
0.01
0.02
0.03unbalancereconstr.
delta2 delta2
delta2
Figure 19: Identification of unbalance positions in the parts of the engine; residuafor the single positions (top), and original and reconstructed unbalance (lower)
f δα,[2], the relative reconstruction error ‖f δα,[2] − f‖/‖f‖ was about 5%. Similarresults hold for the other single unbalances, see Figure 19.
6.1.4 General reconstruction approach
The results of the preceding subsections suggest the following general approachto reconstruct a single unbalance:
1. Do a minimum–norm reconstruction according to subsection 6.1.1 in orderto check the magnitude of the residuum and possible balancing results.
2. Identify the part of the rotor where the unbalance is located choosing thepart for which the residuum equals δ (cf. subsection 6.1.2).
3. Scan through the identified part allowing only single unbalance positions forthe reconstruction. Again choose the position where the residuum equals δ(cf. subsection 6.1.3).
6.2 Reconstruction of an unbalance distribution
To reconstruct an unbalance distribution we have also used the above suggestedapproach. During this section we have chosen an eccentricity of 0.05 mm at thethird disc in the compressor and a run out error of 0.02 mm in the coupling. Wemade a minimum norm reconstruction first in order to get an impression which
28
0 5 10 15 200
0.01
0.02
0.03
0.04
0.05Given unbalance
0 5 102
3
4
5
6Residuum
Iteration number
0 100 200 3000
5
10
15Oscillation in the damper
Frequency in Hz
|h|2
0 100 200 3000
0.05
0.1
0.15
0.2Oscillation after balancing
Frequency in Hz
|h|2
delta2
Figure 20: Reconstruction residuum (lower left) and balancing result (lower rightcompared to upper right) for an unbalance distribution (upper left)
reconstruction and balancing accuracy is possible (Figure 20). In a second stepwe compared the residua of the iteration process restricted to the three singleparts and combinations of two parts of the engine (Figure 21). As a conclusionof Figure 21 we presume our unbalances either in the compressor and the turbineor in the compressor and the coupling. It is not very surprising that we can notidentify only one combination since, for instance, the 7th disc in the turbine isclose to the coupling and a balancing weight there will probably have a satisfyingeffect on the considered frequency interval.This time the third step is parted. We first allow only single potential unbalancepositions and compute the residua (Figure 22). We can see that non of the residuacomes near δ. But when we are looking for a combination of the compressor andone of the other parts, Figure 22 recommends to try the positions of disc 1, 2,3 and 6, since there a decreasing of the residuum is visible. Thus we checkedthe residua for the restricted operators A[i,7], · · · , A[i,10], i = 1, 2, 3, 6 and plottedthe results in Figure 23. Though some of the combinations residua came closeto δ, only the right combination residuum ‖A[3,9] f δα − gδ‖ undercut the δ–leveland the iteration stopped. The other residua remained static at a slightly higherlevel. For the last iterate of the most recommended combinations, we have thefollowing residua (δ2 = 2.68):
‖A[3,9] f δα − gδ‖2 = 2.66
‖A[1,7] f δα − gδ‖2 = 3.87
‖A[2,7] f δα − gδ‖2 = 4.29
‖A[3,8] f δα − gδ‖2 = 3.17
29
0 5 10 150
10
20
30
40
50
60
70
Iteration number
Residua for unbalances at 3rd disc and run out
CompressorTurbineCouplingCompr & turbCompr & coupTurb & coup
delta2
Figure 21: Localization of the parts for an unbalance distribution
0 5 10 1520
30
40
50
60
70
80
Iteration number
Residua for single admitted unbalance positions
1st disc2nd disc3rd disc4th disc5th disc6th disc7th disc8th disc9 run out10 swash
Figure 22: Residua squares for the 10 single potential unbalance positions
30
0 10 20 30 400
10
20
30
40
50
60
70
Iteration number
[1,7][2,7][3,7][6,7]
0 10 20 30 400
10
20
30
40
50
60
70
Iteration number
[1,8][2,8][3,8][6,8]
0 10 20 30 400
20
40
60
80
Iteration number
[1,9][2,9][3,9][6,9]
0 10 20 30 400
10
20
30
40
50
Iteration number
[1,10][2,10][3,10][6,10]
delta2 delta2
delta2 delta2
Figure 23: Residua squares for combinations of two potential unbalance positions;disc 1,2,3 and 6 from the compressor with the remaining positions 7 and 8 fromthe turbine, and 9 and 10 from the coupling
We have to remark that in our case δ is exactly known. For reconstructionswith real data, δ has to be estimated. Therefor it is possible that more thanone combination undercut the estimated δ–level. But due to the regularizationtheory, the residuum for the right combination tends to zero if α tends to zero.As we can see in Figure 23, the wrong combinations remain almost static. Thereconstructed unbalance distribution and the balancing result for A[3,9] can beseen in Figure 24. The relative reconstruction error ‖f δα,[3,9]− f‖/‖f‖ is about 7%.
6.3 Balancing at three fixed positions
Sometimes one is not or not only interested in the identification of an unbalancedistribution but in good balancing results. In practice one often has three fixedpositions where balancing weights can be placed. We assumed disc 1, 5 and 8as such fixed positions and computed the corresponding solution f δα,[1,5,8] and the
balancing result (see Figure 25) for the data generated by the above unbalancedistribution [3, 9]. We can see that the residuum ‖A[1,5,8] f δα,[1,5,8] − gδ‖2 equals δ
much faster than the residuum of the A[3,9]–reconstruction (see Figure 24). Alsothe balancing results are much better. For our three–point balancing we got aremaining oscillation in the squeeze film damper of less than 0.3%.
31
0 5 10 15 20−0.02
0
0.02
0.04
0.06Unbalance and reconstruction
Given unbalanceReconstruction [3,9]
0 5 10 15 200
10
20
30
40
50
Iteration number
Residua for unbalances at 3rd disc and run out
Minim.−norm solution[3,9]
0 100 200 3000
5
10
15Oscillation in the damper
Frequency in Hz
|h|2
0 100 200 3000
0.01
0.02
0.03
0.04Oscillation after balancing
Frequency in Hz|h
|2
Figure 24: Reconstruction results for A[3,9]
0 5 10 15 20−0.02
0
0.02
0.04
0.06Unbalance and reconstruction
Given unbalanceReconstruction [1,5,8]
0 5 10 15 202
4
6
8
Iteration number
Residua for unbalances at 3rd disc and run out
Minim.−norm solution[1,5,8]
0 50 100 150 200 250 3000
5
10
15Oscillation in the damper
Frequency in Hz
|h|2
0 50 100 150 200 250 3000
0.01
0.02
0.03
0.04Oscillation after balancing
Frequency in Hz
|h|2
delta2
Figure 25: Reconstruction results for A[1,5,8]
32
6.4 Data error and reconstruction quality
Surely an important item for the unbalance identification accuracy is the qualityof the available data. Since we did not have access to real data, we had torevert to simulated data. We have used a reference unbalance at the third disc(f(3) = 0.05), produced the corresponding data by forward computation anddisturbed the data with equally distributed random numbers normed to the errorlevel. We varied the error level from 0% to 50%, and applied our three–stepalgorithm. There was no difficulty to identify the engine’s unbalanced parts andfinally the unbalance position. The reconstructed solution frec was computedusing the operator restricted to the 3rd unbalance position A[3]. We defined twomeasures for the accuracy. The first one is the identification accuracy.
Identification accuracy =
(
1 − ‖f − frec‖‖f‖
)
∗ 100%.
Its dependence on the data error level is plotted in the left picture of Figure26. For exact data, the unbalance is exactly reconstructed. The second accuracymeasure considers the balancing effect. We compared the oscillation |h|2 in thesqueeze film damper caused by the reference unbalance with the oscillation afterbalancing with the reconstructed solution |hrec|2 (i.e. forward computation withf − frec). The balancing accuracy was defined by
Balancing accuracy =
(
1 − |hrec|2|h|2
)
∗ 100%.
The right picture of Figure 26 shows the dependence on the data error level. Forexact data we have 100% accuracy, i.e. no remaining unbalance after balancing.For 50% data error we have less than 15% remaining unbalance.
0 10 20 30 40 50 6085
90
95
100
Bal
anci
ng a
ccur
acy
in %
Error level in %0 10 20 30 40 50 60
90
91
92
93
94
95
96
97
98
99
100
Error level in %
Iden
tific
atio
n ac
cura
cy in
%
Figure 26: Identification and balancing accuracy versus noise level
33
7 Summary
We have developed a nonlinear turbine model oriented on the characteristic pa-rameters of a real engine. It has enabled us to compute the oscillation behavior ofthe turbine from given unbalance distributions. Based on this model and on theTikhonov regularization for nonlinear ill–posed problems, we have introduced athree-step algorithm to reconstruct unbalances in the engine from noisy data mea-sured at certain sensor positions on the engine’s casing. For the first time, pointunbalances and discontinuous unbalance distributions could be exactly localizedand reconstructed with satisfying accuracy. Additionally, if balancing positionsare previously fixed, the step algorithm enables us to compute balancing masseswith very good balancing results.The treatment of model errors and a comparison of the linear and the nonlinearmodel were also under investigation and are topics of a forthcoming paper.
34
A Derivative of an inverse matrix
Suppose we want to determine [B(ϕ)−1]′. We use the Taylor expansion of B(ϕ+ε)−1 to determine the derivative from
B(ϕ+ ε)−1 = B(ϕ)−1 + [B(ϕ)−1]′ε+R(ε2).
First we have
B(ϕ+ ε)−1 = [B(ϕ) + [B(ϕ)]′ε+R(ε2)]−1.
Here [B(ϕ)]′ε = Eε is a small term and R(ε2) is ’very small’. We have to deter-mine (B + εE +R(ε2))−1. This can be done by the following chain where we use
the Neumann serie (Id− A)−1 =∞∑
k=0
Ak.
(B + εE +R(ε2))−1 = [B(Id− (−εB−1E) +B−1R(ε2))]−1
= [Id− (−εB−1E) +R1(ε2)]−1B−1
=∞∑
k=0
(−εB−1E +R1(ε2))kB−1
= B−1 − εB−1EB−1 +R2(ε2).
It follows that
B(ϕ+ ε)−1 = B(ϕ)−1 − εB(ϕ)−1[B(ϕ)]′B(ϕ)−1 +R2(ε2),
hence
[B(ϕ)−1]′ = −B(ϕ)−1[B(ϕ)]′B(ϕ)−1.
B
We want to proof
(Id+ 〈q, ·〉w)−1 = Id− 1
1 + 〈q, w〉〈q, ·〉w.
Since 〈q, ·〉w = wqt and wqtwqt = wqt〈q, w〉 we have
(Id+ wqt)(Id− cwqt) = Id+ wqt − cwqt − cwqtwqt
= Id+ wqt(1 − c− c〈q, w〉)= Id
⇔ 1 − c− c〈q, w〉 = 0
⇔ c =1
1 + 〈q, w〉 .
35
References
[1] R. D. Blevins. Formulas for Natural Frequency and Mode Shape. van Nos-trand Reinhold, New York, 1979.
[2] V. Dicken. Inverse Imbalance Reconstruction in Rotordynamics. CERESreport on the ZeTeM subcontract for Task 6.2., 2001.
[3] H. W. Engl, M. Hanke, and A. Neubauer. Regularization of Inverse Problems.Kluwer, Dordrecht, 1996.
[4] R. Gasch and K. Knothe. Strukturdynamik Bd.2 Kontinua und ihreDiskretisierung. Springer, Berlin, 1989.
[5] R. Gasch, R. Nordmann, and H. Pfutzner. Rotordynamik. Springer, Berlin,2002.
[6] R. Holmes. Rotor vibration control using squeeze film dampers. IFToMM,Fifth International Conference on Rotordynamics, 1998.
[7] K. Knothe and H. Wessls. Finite Elemente - eine Einfuhrung fur Ingenieure.Springer, Berlin, 1992.
[8] A. K. Louis. Inverse und schlecht gestellte Probleme. Teubner, Stuttgart,1989.
[9] R. Ramlau. Morozov’s discrepancy principle for Tikhonov regularization ofnonlinear operators. Numer. Funct. Anal. and Optimiz, 23(1&2):147–172,2002.
[10] R. Ramlau. A steepest descent algorithm for the global minimization of theTikhonov– functional. Inverse Problems, 18(2):381–405, 2002.
[11] R. Ramlau. TIGRA–an iterative algorithm for regularizing nonlinear ill–posed problems. Inverse Problems, 19(2):433–467, 2003.
[12] A. Rienacker, D. Peters, G. Tokar, and B. Domes. Einsatz inverser Metho-den zur Bestimmung der Rotorunwuchtverteilung eines Flugtriebwerks. VDIBerichte: Schwingungsuberwachung und -diagnose von Maschinen und An-lagen: Tagung Frankenthal, 27./28. Mai 1999, (1466):363–373, 1999.
[13] O. Scherzer. The use of Morozov’s discrepancy principle for Tikhonov regu-larization for solving nonlinear ill–posed problems. Computing, (51):45–60,1993.
36
Berichte aus der Technomathematik ISSN 1435-7968
http://www.math.uni-bremen.de/zetem/berichte.html
— Vertrieb durch den Autor —
Reports Stand: 24. Juli 2003
98–01. Peter Benner, Heike Faßbender:An Implicitly Restarted Symplectic Lanczos Method for the Symplectic Eigenvalue Problem,Juli 1998.
98–02. Heike Faßbender:Sliding Window Schemes for Discrete Least-Squares Approximation by Trigonometric Poly-nomials, Juli 1998.
98–03. Peter Benner, Maribel Castillo, Enrique S. Quintana-Ortı:Parallel Partial Stabilizing Algorithms for Large Linear Control Systems, Juli 1998.
98–04. Peter Benner:Computational Methods for Linear–Quadratic Optimization, August 1998.
98–05. Peter Benner, Ralph Byers, Enrique S. Quintana-Ortı, Gregorio Quintana-Ortı:Solving Algebraic Riccati Equations on Parallel Computers Using Newton’s Method withExact Line Search, August 1998.
98–06. Lars Grune, Fabian Wirth:On the rate of convergence of infinite horizon discounted optimal value functions, November1998.
98–07. Peter Benner, Volker Mehrmann, Hongguo Xu:A Note on the Numerical Solution of Complex Hamiltonian and Skew-Hamiltonian Eigen-value Problems, November 1998.
98–08. Eberhard Bansch, Burkhard Hohn:Numerical simulation of a silicon floating zone with a free capillary surface, Dezember 1998.
99–01. Heike Faßbender:The Parameterized SR Algorithm for Symplectic (Butterfly) Matrices, Februar 1999.
99–02. Heike Faßbender:Error Analysis of the symplectic Lanczos Method for the symplectic Eigenvalue Problem,Marz 1999.
99–03. Eberhard Bansch, Alfred Schmidt:Simulation of dendritic crystal growth with thermal convection, Marz 1999.
99–04. Eberhard Bansch:Finite element discretization of the Navier-Stokes equations with a free capillary surface,Marz 1999.
99–05. Peter Benner:Mathematik in der Berufspraxis, Juli 1999.
99–06. Andrew D.B. Paice, Fabian R. Wirth:Robustness of nonlinear systems and their domains of attraction, August 1999.
99–07. Peter Benner, Enrique S. Quintana-Ortı, Gregorio Quintana-Ortı:Balanced Truncation Model Reduction of Large-Scale Dense Systems on Parallel Comput-ers, September 1999.
99–08. Ronald Stover:Collocation methods for solving linear differential-algebraic boundary value problems, Septem-ber 1999.
99–09. Huseyin Akcay:Modelling with Orthonormal Basis Functions, September 1999.
99–10. Heike Faßbender, D. Steven Mackey, Niloufer Mackey:Hamilton and Jacobi come full circle: Jacobi algorithms for structured Hamiltonian eigen-problems, Oktober 1999.
99–11. Peter Benner, Vincente Hernandez, Antonio Pastor:On the Kleinman Iteration for Nonstabilizable System, Oktober 1999.
99–12. Peter Benner, Heike Faßbender:A Hybrid Method for the Numerical Solution of Discrete-Time Algebraic Riccati Equations,November 1999.
99–13. Peter Benner, Enrique S. Quintana-Ortı, Gregorio Quintana-Ortı:Numerical Solution of Schur Stable Linear Matrix Equations on Multicomputers, November1999.
99–14. Eberhard Bansch, Karol Mikula:Adaptivity in 3D Image Processing, Dezember 1999.
00–01. Peter Benner, Volker Mehrmann, Hongguo Xu:Perturbation Analysis for the Eigenvalue Problem of a Formal Product of Matrices, Januar2000.
00–02. Ziping Huang:Finite Element Method for Mixed Problems with Penalty, Januar 2000.
00–03. Gianfrancesco Martinico:Recursive mesh refinement in 3D, Februar 2000.
00–04. Eberhard Bansch, Christoph Egbers, Oliver Meincke, Nicoleta Scurtu:Taylor-Couette System with Asymmetric Boundary Conditions, Februar 2000.
00–05. Peter Benner:Symplectic Balancing of Hamiltonian Matrices, Februar 2000.
00–06. Fabio Camilli, Lars Grune, Fabian Wirth:A regularization of Zubov’s equation for robust domains of attraction, Marz 2000.
00–07. Michael Wolff, Eberhard Bansch, Michael Bohm, Dominic Davis:Modellierung der Abkuhlung von Stahlbrammen, Marz 2000.
00–08. Stephan Dahlke, Peter Maaß, Gerd Teschke:Interpolating Scaling Functions with Duals, April 2000.
00–09. Jochen Behrens, Fabian Wirth:A globalization procedure for locally stabilizing controllers, Mai 2000.
00–10. Peter Maaß, Gerd Teschke, Werner Willmann, Gunter Wollmann:Detection and Classification of Material Attributes – A Practical Application of WaveletAnalysis, Mai 2000.
00–11. Stefan Boschert, Alfred Schmidt, Kunibert G. Siebert, Eberhard Bansch, Klaus-WernerBenz, Gerhard Dziuk, Thomas Kaiser:Simulation of Industrial Crystal Growth by the Vertical Bridgman Method, Mai 2000.
00–12. Volker Lehmann, Gerd Teschke:Wavelet Based Methods for Improved Wind Profiler Signal Processing, Mai 2000.
00–13. Stephan Dahlke, Peter Maass:A Note on Interpolating Scaling Functions, August 2000.
00–14. Ronny Ramlau, Rolf Clackdoyle, Frederic Noo, Girish Bal:Accurate Attenuation Correction in SPECT Imaging using Optimization of Bilinear Func-tions and Assuming an Unknown Spatially-Varying Attenuation Distribution, September2000.
00–15. Peter Kunkel, Ronald Stover:Symmetric collocation methods for linear differential-algebraic boundary value problems,September 2000.
00–16. Fabian Wirth:The generalized spectral radius and extremal norms, Oktober 2000.
00–17. Frank Stenger, Ahmad Reza Naghsh-Nilchi, Jenny Niebsch, Ronny Ramlau:A unified approach to the approximate solution of PDE, November 2000.
00–18. Peter Benner, Enrique S. Quintana-Ortı, Gregorio Quintana-Ortı:Parallel algorithms for model reduction of discrete–time systems, Dezember 2000.
00–19. Ronny Ramlau:A steepest descent algorithm for the global minimization of Tikhonov–Phillips functional,Dezember 2000.
01–01. Efficient methods in hyperthermia treatment planning:Torsten Kohler, Peter Maass, Peter Wust, Martin Seebass, Januar 2001.
01–02. Parallel Algorithms for LQ Optimal Control of Discrete-Time Periodic Linear Systems:Peter Benner, Ralph Byers, Rafael Mayo, Enrique S. Quintana-Ortı, Vicente Hernandez,Februar 2001.
01–03. Peter Benner, Enrique S. Quintana-Ortı, Gregorio Quintana-Ortı:Efficient Numerical Algorithms for Balanced Stochastic Truncation, Marz 2001.
01–04. Peter Benner, Maribel Castillo, Enrique S. Quintana-Ortı:Partial Stabilization of Large-Scale Discrete-Time Linear Control Systems, Marz 2001.
01–05. Stephan Dahlke:Besov Regularity for Edge Singularities in Polyhedral Domains, Mai 2001.
01–06. Fabian Wirth:A linearization principle for robustness with respect to time-varying perturbations, Mai2001.
01–07. Stephan Dahlke, Wolfgang Dahmen, Karsten Urban:Adaptive Wavelet Methods for Saddle Point Problems - Optimal Convergence Rates, Juli2001.
01–08. Ronny Ramlau:Morozov’s Discrepancy Principle for Tikhonov regularization of nonlinear operators, Juli2001.
01–09. Michael Wolff:Einfuhrung des Drucks fur die instationaren Stokes–Gleichungen mittels der Methode vonKaplan, Juli 2001.
01–10. Stephan Dahlke, Peter Maaß, Gerd Teschke:Reconstruction of Reflectivity Desities by Wavelet Transforms, August 2001.
01–11. Stephan Dahlke:Besov Regularity for the Neumann Problem, August 2001.
01–12. Bernard Haasdonk, Mario Ohlberger, Martin Rumpf, Alfred Schmidt, Kunibert G.Siebert:h–p–Multiresolution Visualization of Adaptive Finite Element Simulations, Oktober 2001.
01–13. Stephan Dahlke, Gabriele Steidl, Gerd Teschke:Coorbit Spaces and Banach Frames on Homogeneous Spaces with Applications to AnalyzingFunctions on Spheres, August 2001.
02–01. Michael Wolff, Michael Bohm:Zur Modellierung der Thermoelasto-Plastizitat mit Phasenumwandlungen bei Stahlen sowieder Umwandlungsplastizitat, Februar 2002.
02–02. Stephan Dahlke, Peter Maaß:An Outline of Adaptive Wavelet Galerkin Methods for Tikhonov Regularization of InverseParabolic Problems, April 2002.
02–03. Alfred Schmidt:A Multi-Mesh Finite Element Method for Phase Field Simulations, April 2002.
02–04. Sergey N. Dachkovski, Michael Bohm:A Note on Finite Thermoplasticity with Phase Changes, Juli 2002.
02–05. Michael Wolff, Michael Bohm:Phasenumwandlungen und Umwandlungsplastizitat bei Stahlen im Konzept der Thermoelasto-Plastizitat, Juli 2002.
02–06. Gerd Teschke:Construction of Generalized Uncertainty Principles and Wavelets in Anisotropic SobolevSpaces, August 2002.
02–07. Ronny Ramlau:TIGRA – an iterative algorithm for regularizing nonlinear ill–posed problems, August 2002.
02–08. Michael Lukaschewitsch, Peter Maaß, Michael Pidcock:Tikhonov regularization for Electrical Impedance Tomography on unbounded domains, Ok-tober 2002.
02–09. Volker Dicken, Peter Maaß, Ingo Menz, Jenny Niebsch, Ronny Ramlau:Inverse Unwuchtidentifikation an Flugtriebwerken mit Quetscholdampfern, Oktober 2002.
02–10. Torsten Kohler, Peter Maaß, Jan Kalden:Time-series forecasting for total volume data and charge back data, November 2002.
02–11. Angelika Bunse-Gerstner:A Short Introduction to Iterative Methods for Large Linear Systems, November 2002.
02–12. Peter Kunkel, Volker Mehrmann, Ronald Stover:Symmetric Collocation for Unstructured Nonlinear Differential-Algebraic Equations of Ar-bitrary Index, November 2002.
02–13. Michael Wolff:Ringvorlesung: Distortion Engineering 2Kontinuumsmechanische Modellierung des Materialverhaltens von Stahl unter Berucksich-tigung von Phasenumwandlungen, Dezember 2002.
02–14. Michael Bohm, Martin Hunkel, Alfred Schmidt, Michael Wolff:Evaluation of various phase-transition models for 100Cr6 for application in commercialFEM programs, Dezember 2002.
03–01. Michael Wolff, Michael Bohm, Serguei Dachkovski:Volumenanteile versus Massenanteile - der Dilatometerversuch aus der Sicht der Kontinu-umsmechanik, Januar 2003.
03–02. Daniel Kessler, Ricardo H. Nochetto, Alfred Schmidt:A posteriori error control for the Allen-Cahn Problem: circumventing Gronwall’s inequality,Marz 2003.
03–03. Michael Bohm, Jorg Kropp, Adrian Muntean:On a Prediction Model for Concrete Carbonation based on Moving Interfaces - Interfaceconcentrated Reactions, April 2003.
03–04. Michael Bohm, Jorg Kropp, Adrian Muntean:A Two-Reaction-Zones Moving-Interface Model for Predicting Ca(OH)2 Carbonation inConcrete, April 2003.
03–05. Vladimir L. Kharitonov, Diederich Hinrichsen:Exponential estimates for time delay systems, May 2003.
03–06. Michael Wolff, Michael Bohm, Serguei Dachkovski, Gunther Lowisch:Zur makroskopischen Modellierung von spannungsabhangigem Umwandlungsverhalten undUmwandlungsplastizitat bei Stahlen und ihrer experimentellen Untersuchung in einfachenVersuchen, Juli 2003.
03–07. Serguei Dachkovski, Michael Bohm, Alfred Schmidt, Michael Wolff:Comparison of several kinetic equations for pearlite transformation in 100Cr6 steel,Juli 2003.
03–08. Volker Dicken, Peter Maass, Ingo Menz, Jenny Niebsch, Ronny Ramlau:Nonlinear Inverse Unbalance Reconstruction in Rotor dynamics,Juli 2003.